Frequency sensing systems and methods

ABSTRACT

Systems and methods may be used to measure a frequency of a power delivery system and/or of a supply signal transmitted to a load. A system may record an input waveform, determine a frequency of the input waveform at a present time based at least in part on the input waveform and a derivative of the input waveform, and control an operation of a power delivery system based at least in part on the determined frequency.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 62/734,900, filed on Sep. 21, 2018 and entitled “Electric PowerDelivery System Frequency Measurement,” which is incorporated byreference in its entirety for all purposes.

BACKGROUND

This disclosure relates to sensing operations in a power deliverysystem. More particularly, this disclosure relates to frequency sensingoperations of an electrical waveform based at least in part on aderivative calculation of the electrical waveform.

This section is intended to introduce the reader to various aspects ofart that may be related to various aspects of the present techniques,which are described and/or claimed below. This discussion is believed tobe helpful in providing the reader with background information tofacilitate a better understanding of the various aspects of the presentdisclosure. Accordingly, it should be understood that these statementsare to be read in this light, and not as admissions of any kind.

Electric power delivery systems are widely used to generate anddistribute electric power to loads. While some systems (or portions ofsystems) operate in direct current (DC), many electric power deliverysystems operate (or have portions that operate) in alternating current(AC). In AC systems, the power flowing through the conductors and otherpower system equipment is from current waveforms and voltage waveformsalternating between high and low peaks generally in a sinusoidalfashion. The frequency of the alternating waveforms are a valuable powersystem measurement for system control and protection, and for many othermonitoring and protection functions. For example, frequency measurementoperations, load shedding operations, overexcitation protectionoperations, synchrophasor measurement operations, switching operations,bus transfer operations, and so on may each use a measured frequencyduring operations. Some measurements of power system frequency, however,may be inefficient and/or take a complete period of the power systemfrequency to determine.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the presentdisclosure will become better understood when the following detaileddescription is read with reference to the accompanying drawings in whichlike characters represent like parts throughout the drawings, wherein:

FIG. 1 illustrates a block diagram of an embodiment of a power delivery(e.g., transmission) system, in accordance with an embodiment;

FIG. 2 illustrates a block diagram of a protection system of the powerdelivery system of FIG. 1, in accordance with an embodiment;

FIG. 3 is a flow diagram of a process for controlling the power deliverysystem of FIG. 1 and/or the protection system of FIG. 2 based at leastin part on a determined frequency of a waveform (e.g., voltage waveform,current waveform), in accordance with an embodiment;

FIG. 4A is a graph of results of a first frequency measuring method, inaccordance with an embodiment;

FIG. 4B is a graph of results of a second frequency measuring method, inaccordance with an embodiment;

FIG. 5 is a flow diagram of a process for using the second frequencymeasuring method to determine the frequency of the voltage for use inthe process of FIG. 3, in accordance with an embodiment;

FIG. 6 is a graph of a first operation of the process of FIG. 5, inaccordance with an embodiment,

FIG. 7 is a graph of a second operation of the process of FIG. 5, inaccordance with an embodiment;

FIG. 8 is a graph of a third operation of the process of FIG. 5, inaccordance with an embodiment;

FIG. 9 is a graph of a fourth operation of the process of FIG. 5, inaccordance with an embodiment;

FIG. 10A is a graph of an example operation performed via the process ofFIG. 5, in accordance with an embodiment;

FIG. 10B is a graph of another example operation performed via theprocess of FIG. 5, in accordance with an embodiment;

FIG. 11 is a graph of the determined frequency of a simulated voltagewaveform determined via the process of FIG. 5, in accordance with anembodiment; and

FIG. 12 is a graph of an accuracy of the frequency of the voltagedetermined via the process of FIG. 5, in accordance with an embodiment.

DETAILED DESCRIPTION

One or more specific embodiments will be described below. In an effortto provide a concise description of these embodiments, not all featuresof an actual implementation are described in the specification. Itshould be appreciated that in the development of any such actualimplementation, as in any engineering or design project, numerousimplementation-specific decisions must be made to achieve thedevelopers' specific goals, such as compliance with system-related andbusiness-related constraints, which may vary from one implementation toanother. Moreover, it should be appreciated that such a developmenteffort might be complex and time consuming, but would nevertheless be aroutine undertaking of design, fabrication, and manufacture for those ofordinary skill having the benefit of this disclosure. Certain examplescommensurate in scope with the originally claimed subject matter arediscussed below. These examples are not intended to limit the scope ofthe disclosure. Indeed, the present disclosure may encompass a varietyof forms that may be similar to or different from the examples set forthbelow.

When introducing elements of various embodiments of the presentdisclosure, the articles “a,” “an,” and “the” are intended to mean thatthere are one or more of the elements. The terms “comprising,”“including,” and “having” are intended to be inclusive and mean thatthere may be additional elements other than the listed elements.Additionally, it should be understood that references to “oneembodiment” or “an embodiment” of the present disclosure are notintended to be interpreted as excluding the existence of additionalembodiments that also incorporate the recited features. Furthermore, thephrase A “based on” B is intended to mean that A is at least partiallybased on B. Moreover, unless expressly stated otherwise, the term “or”is intended to be inclusive (e.g., logical OR) and not exclusive (e.g.,logical XOR). In other words, the phrase A “or” B is intended to mean A,B, or both A and B.

Electric power delivery systems may generate and distribute electricpower to loads. In some electric power delivery systems, at least aportion of the electric power delivery system operates in direct current(DC). However, many electric power delivery systems operate at leastpartially in alternating current (AC). In AC systems, operations of theelectric power delivery system are sometimes based on a frequency of thewaveforms generated and/or distributed via the electric power deliverysystem. These operations may include, for example, frequency controloperations, frequency protection operations, load shedding operations,overexcitation protection operations, synchrophasor measurementoperations, switching operations, bus transfer operations, or the like.Accordingly, accurate measurement of power system frequency may beuseful for proper monitoring and protection of the power system.

Since voltage waveforms and/or current waveforms of the electric powerdelivery system are periodic, frequency may be measured using the periodof the waveform. For example, a frequency may be determined by measuringa time between consecutive high peaks, a time between consecutive lowpeaks, a time between rising zero crossings, a time between falling zerocrossings, or the like. However, such frequency calculations may use acomplete power system cycle. In many applications, it may be desirableto calculate frequency in less than a power system cycle. Indeed,although ideal waveforms are periodic in steady state, power systemfrequencies may fluctuate. In a functioning power system, voltagewaveforms and/or current waveforms are not precisely periodic, and thefrequency may vary within periods. Thus, a system able to determinepower system frequency in less than a complete power system cycle usingthe available waveforms such as current, voltage, and the like, may bedesired.

As described below, a system may use a waveform (e.g., voltage, current)and a derivative of the waveform to determine a frequency of thewaveform in less than a complete power system cycle. By using thesystems and methods described herein, a frequency determination may bemore efficient, and thus may improve a speed of frequency determination,as well as the speed of subsequent control operations that use thedetermined frequency.

The embodiments of the disclosure will be best understood by referenceto the drawings, wherein like parts are designated by like numeralsthroughout. It will be readily understood that the components of thedisclosed embodiments, as generally described and illustrated in thefigures herein, could be arranged and designed in a wide variety ofdifferent configurations. Thus, the following detailed description ofthe embodiments of the systems and methods of the disclosure is notintended to limit the scope of the disclosure, as claimed, but is merelyrepresentative of possible embodiments of the disclosure. In addition,the steps of a method do not necessarily need to be executed in anyspecific order, or even sequentially, nor need the steps be executedonly once, unless otherwise specified.

In some cases, well-known features, structures or operations are notshown or described in detail. Furthermore, the described features,structures, or operations may be combined in any suitable manner in oneor more embodiments. It will also be readily understood that thecomponents of the embodiments as generally described and illustrated inthe figures herein could be arranged and designed in a wide variety ofdifferent configurations.

Several aspects of the embodiments described may be implemented assoftware modules or components. As used herein, a software module orcomponent may include any type of computer instruction or computerexecutable code located within a memory device and/or transmitted aselectronic signals over a system bus or wired or wireless network. Asoftware module or component may, for instance, comprise one or morephysical or logical blocks of computer instructions, which may beorganized as a routine, program, object, component, data structure, orthe like, that performs one or more tasks or implements particularabstract data types.

In certain embodiments, a particular software module or component maycomprise disparate instructions stored in different locations of amemory device, which together implement the described functionality ofthe module. Indeed, a module or component may comprise a singleinstruction or many instructions, and may be distributed over severaldifferent code segments, among different programs, and across severalmemory devices. Some embodiments may be practiced in a distributedcomputing environment where tasks are performed by a remote processingdevice linked through a communications network. In a distributedcomputing environment, software modules or components may be located inlocal and/or remote memory storage devices. In addition, data being tiedor rendered together in a database record may be resident in the samememory device, or across several memory devices, and may be linkedtogether in fields of a record in a database across a network.

Embodiments may be provided as a computer program product including anon-transitory computer and/or machine-readable medium having storedthereon instructions that may be used to program a computer (or otherelectronic device) to perform processes described herein. For example, anon-transitory computer-readable medium may store instructions that,when executed by a processor of a computer system, cause the processorto perform certain methods disclosed herein.

The non-transitory computer-readable medium may include, but is notlimited to, hard drives, floppy diskettes, optical disks, compact discread-only memories (CD-ROMs), digital versatile disc read-only memories(DVD-ROMs), read-only memories (ROMs), random access memories (RAMs),erasable programmable read-only memories (EPROMs), electrically erasableprogrammable read-only memories (EEPROMs), magnetic or optical cards,solid-state memory devices, or other types of machine-readable mediasuitable for storing electronic and/or processor executableinstructions.

FIG. 1 illustrates a block diagram of a power delivery system 20 (e.g.,electrical power transmission system). The power delivery system 20 maybe any suitable power delivery system 20, such as a three-phase powerdelivery system. Moreover, although the one-line block diagram is usedas a simplified example, the systems and methods disclosed herein may beused in conjunction with any suitable electric power delivery system,such as a power transmission system, a power distribution system, radialpower distribution systems, bi-directional power systems, or the like.The power delivery system 20 may generate, transmit, and distributeelectric energy to loads. The power delivery system 20 may includevarious types of equipment, such as electric generators, powertransformers, power transmission and/or delivery lines, circuitbreakers, busses, loads, and the like. A variety of other types ofequipment may also be included in power delivery system 20, such asvoltage regulators, capacitor banks, and the like. Furthermore, thesystem and methods disclosed herein may additionally or alternatively beused with loads as well, such as to determine a frequency of a voltagesupplied to a load without considering a frequency of the system as awhole. In the illustrated example, the power delivery system 20 includesa power line 22 that transfers electrical energy from a first powergenerator 24 (e.g., local power generator) and a second power generator26 (e.g., remote power generator) to one or more loads.

The power delivery system 20 may be monitored, controlled, automated,and/or protected using protection systems 30 and 32. The protectionsystems 30 and 32 may each include one or more intelligent electronicdevices (IEDs), such as a local relay 40 and a remote relay 50. Forexample, the IEDs may be used to monitor equipment of many types,including electric transmission lines, electric distribution lines,current transformers, busses, switches, circuit breakers, reclosers,transformers, autotransformers, tap changers, voltage regulators,capacitor banks, generators, motors, pumps, compressors, valves, and avariety of other types of monitored equipment. Note that, as usedherein, the local relay 40 may refer to the relay that is determiningthe location of the fault as a distance from that relay. Further, theremote relay 50 may refer to the relay that transmits data (e.g.,current measurements and voltage measurements) to be used by the localrelay 40 in determining the location of the fault. The remote relay 50may be any suitable distance from the local relay 40.

As used herein, an IED (such as the local relay 40 and the remote relay50) may refer to any microprocessor-based device that monitors,controls, automates, and/or protects monitored equipment within thepower delivery system 20. Such devices may include, for example, remoteterminal units, differential relays, distance relays, directionalrelays, feeder relays, overcurrent relays, voltage regulator controls,voltage relays, breaker failure relays, generator relays, motor relays,automation controllers, bay controllers, meters, recloser controls,communications processors, computing platforms, programmable logiccontrollers (PLCs), programmable automation controllers, input andoutput modules, and the like. The term IED may be used to describe anindividual IED or a system that includes multiple IEDs.

A common time signal may be distributed throughout the power deliverysystem 20. Utilizing a common or universal time source may enable theIEDs to generate time synchronized data. In various embodiments, relays40 and 50 may receive the common time signal. The common time signal maybe distributed in the power delivery system 20 using a communicationsnetwork or using a common time source, such as a Global NavigationSatellite System (“GNSS”), or the like.

According to various embodiments, the local relay 40 and the remoterelay 50 may use communication circuitry 56 and 58 to communicate witheach other, with one or more other IEDs 70, and/or with a centralmonitoring station 72. In some embodiments, the local relay 40 and theremote relay 50 may communicate with the IED 70 and/or the centralmonitoring station 72 directly or via a communication network. Thecentral monitoring station 72 may include one or more of a variety oftypes of systems. For example, central monitoring station 72 may includea supervisory control and data acquisition (SCADA) system and/or a widearea control and situational awareness (WACSA) system. The local relay40 and the remote relay 50 may communicate over various media such asdirect communication or over a wide-area communications network.

Network communication may be facilitated by networking devicesincluding, but not limited to, multiplexers, access points, routers,hubs, gateways, firewalls, and switches. In some embodiments, IEDs andnetwork devices may include physically distinct devices. In otherembodiments, IEDs and network devices may be composite devices, or maybe configured in a variety of ways to perform overlapping functions IEDsand network devices may include multi-function hardware (e.g.,processors, computer-readable storage media, communications interfaces,etc.) that may be utilized to perform a variety of tasks that pertain tonetwork communications and/or to operation of equipment within the powerdelivery system 20.

As explained below, the local relay 40 and/or the remote relay 50 maymonitor the electrical characteristics of power on the power line 22 viasensor circuitry 42 and 52. Each of the local relay 40 and the remoterelay 50 may be communicatively coupled to a respective circuit breaker44 and 54. Upon occurrence of a fault 80, the local relay 40, the remoterelay 50, the other IED 70, and/or the central monitoring station 72,may effect a control operation on the power delivery system 20, such asopening the local circuit breaker 44 or opening the remote circuitbreaker 54.

FIG. 2 illustrates a block diagram of the protection system 30 that maybe used to determine a location of a fault, determine a frequency of avoltage, effect a control operation on the power delivery system 20, orthe like. In the illustrated embodiment, the protection system 30includes the local relay 40, the sensor circuitry 42, and the localcircuit breaker 44. The local relay 40 may include a bus 100 connectinga processor 102 or processing unit(s) to a memory 104, acomputer-readable storage medium 106, input circuitry 108, protectioncircuitry 110, and one or more displays 112. In some embodiments, theprocessor 102 may include two or more processors. The computer-readablestorage medium 106 may include and/or interface with software, hardware,and/or firmware modules for implementing various portions of the systemsand methods described herein. The computer-readable storage medium 106may be the repository of one or more modules and/or executableinstructions to implement any of the processes described herein. In someembodiments, the computer-readable storage medium 106 and the modulestherein may all be implemented as hardware components, such as viadiscrete electrical components, via a Field Programmable Gate Array(“FPGA”), and/or via one or more Application Specific IntegratedCircuits (“ASICs”).

The processor 102 may process inputs received via the input circuitry108 and/or the communication circuitry 56. The processor 102 may operateusing any number of processing rates and architectures. The processor102 may perform various algorithms and calculations described hereinusing computer-executable instructions stored on computer-readablestorage medium 106. In some embodiments, the processor 102 may beembodied as a microprocessor, a general purpose integrated circuit, anASIC, a FPGA, and/or other programmable logic devices.

The sensor circuitry 42 may include a current transformer 130 and/or apotential (e.g., voltage) transformer 132. The input circuitry 108 mayreceive electric current waveforms and/or voltage waveforms from thecurrent transformer 130 and the potential transformer 132 respectively,transform the signals using respective potential transformer(s) 134 and136 to a level that may be sampled, and sample the signals using, forexample, analog-to-digital (A/D) converter(s) 118 to produce digitalsignals representative of measured current and measured voltage on thepower line. Similar values may also be received from other distributedcontrollers, station controllers, regional controllers, or centralizedcontrollers. The values may be in a digital format or other format. Incertain embodiments, the input circuitry 108 may be utilized to monitorcurrent signals associated with a portion of a power delivery system 20.Further, the input circuitry 108 may monitor a wide range ofcharacteristics associated with monitored equipment, including equipmentstatus, temperature, frequency, pressure, density, infrared absorption,radio-frequency information, partial pressures, viscosity, speed,rotational velocity, mass, switch status, valve status, circuit breakerstatus, tap status, meter readings, conductor sag, and the like.

The A/D converter 118 may be connected to the processor 102 by way ofthe bus 100, through which digitized representations of current andvoltage waveforms may be transmitted to the processor 102. As describedabove, the processor 102 may be used to monitor and/or protect portionsof the power delivery system 20, and issue control instructions inresponse to the same (e.g., instructions implementing protectiveactions). For example, the processor 102 may determine a location of afault on a power line 22 based on the digitized representations of thecurrent and/or voltage waveforms.

In response to detecting a fault or otherwise abnormal behavior, theprocessor 102 may toggle a control operation on the power deliverysystem 20 via the protection circuitry 110. For example, the processor102 may send a signal to control operation of the local circuit breaker44, such as to disconnect the power line 22 from the local powergenerator 24. As illustrated, the local relay 40 may include the display112 to display alarms indicating the location of the fault to anoperator. The communication circuitry 56 may include a transceiver tocommunicate with one or more other IEDs and/or a central monitoringstation, or the like. In some embodiments, the processor 102 may causethe transceiver to send a signal indicating the location of the fault.For example, the processor 102 may send, via the transceiver of thecommunication circuitry 56, a signal indicating the location of thefault to the central monitoring station 72. Further, the processor 102may activate the alarms upon detection of the fault.

FIG. 3 is a flow diagram of a process 150 for operating the powerdelivery system 20, the protection circuitry 110, loads, or the likebased at least in part on a determined frequency of a waveform (e.g.,voltage, current). The relay 40 may perform the process 150 (e.g.,frequency determination operations) using, for example, the processor102. However, it should be understood that, although the relay 40 mayperform the determination locally, the relay 40 may transmit thedetermined frequency through a communication network to other controlcircuitry. For example, the frequency may be sent to a supervisorycontrol and data acquisition (SCADA) system for use in additional ordifferent control operations. The frequency may be determined viacomputations involving signal derivatives. The process 150 is describedbelow as being performed by the processor 102; however, it should beunderstood that any suitable processor or processing circuitry mayperform the process 150 to perform a frequency sensing operation.Furthermore, although depicted in a particular order, it should beunderstood that any suitable order, and fewer or additional operations,may be performed when performing the process 150. Additionally, theprocess 150 is described below as based on recorded voltage datacorresponding to a voltage waveform of the power delivery system 20. Itis noted that in some cases recorded current data may be used todetermine the frequency.

For the example of a voltage waveform, at block 152, the processor 102may record sensed voltage data over time. The processor 102 may becommunicative coupled to sensing circuitry, such as the potentialtransformer 132 or the current transformer 130. Sensing data may bereceived by the processor 102 via the input circuitry 108, thecommunication circuitry 56, or the like. The sensing data may indicateto the processor 102 a voltage at a time of sensing. The processor 102may receive the sensing data and store the sensing data in memory 104,such as to generate a historic sensing data record. The sensing data mayalso be stored with a timestamp defining the time of sensing of thedata. The timestamp may be assigned to the sensing data based at leastin part on a time received from a common time source.

At block 154, the processor 102 may determine a frequency of the voltagewaveform sensed at a present time (e.g., most recent time). Thederivative of the voltage waveform may be computed numerically from thesensing data samples recorded over time. Operations performed by theprocessor 102 to compute the derivative of the voltage waveform and thefrequency of the voltage waveform are further explained via a methoddepicted in FIG. 5. Generally, the operations enable the processor 102to compare a present value of the voltage waveform (e.g., a presentlysensed voltage value, sensing data) with one or more previous values ofthe derivatives of the voltage waveform to determine when the derivativeof the voltage waveform equals the present value of the voltagewaveform, thus identifying the frequency of the voltage waveform. Insome embodiments, this comparison is a discrete comparison, wheredistinctly sensed voltage values are recorded and compared over time.Furthermore, the processor 102 may perform a discrete derivativedetermination on the sensed data of the voltage waveform, where eachsensed data received by the processor 102 may be associated with arespectively determined derivative value. In this way, in some cases,the sensed voltage values may not be curve-fitted into a sinusoidalwaveform or other definition, and the comparison is performed on thediscretely-obtained sensed voltage values.

At block 156, the processor 102 may control an operation of a loadand/or a power system based at least in part on the determinedfrequency. In this way, the processor 102 may be a part of a controlloop that responds in real-time to operating condition changes. Forexample, if the determined frequency of the voltage exceeds a threshold,the processor 102 may generate control signals to slow a rotation of itsload. As a second example, synchrophasor operation may use frequencyalignment and determinations when selecting when to actuate a relayand/or circuit breaker. Thus, synchrophasor operation may also improvefrom an improved frequency determination process.

FIG. 4A is a graph 168 comparing an actual frequency of a simulatedvoltage waveform and a determined frequency of the simulated voltagewaveform. FIG. 4B is a graph 174 comparing an actual frequency of thesimulated voltage waveform and a determined frequency of the simulatedvoltage waveform, where the determined frequency is determined based atleast in part on a derivative of the simulated voltage waveform. Forease of explanation, FIG. 4A and FIG. 4B are explained together. Thegraph 168 depicts a tracking of a first frequency determination methodby comparing frequency (ordinate 170) to time (abscissa 172) while thegraph 174 depicts a tracking of a second frequency determination methodby comparing frequency (ordinate 170) to time (abscissa 172). The secondfrequency determination method is described further via FIG. 5 andimproves on tracking of the measured frequency (e.g., line 176) to theactual frequency (e.g., line 178) shown via the graph 168. By improvingthe tracking of the measured frequency, the second frequencydetermination method may improve accuracy and response times ofcontrollers that use the second frequency determination method tocalculate frequency of a voltage or current signal during operation.

Keeping this in mind, FIG. 5 is a flow diagram of a process 190 forusing the second frequency measuring method to determine the frequencyof the voltage for use in the process 150. The process 190 is describedbelow as being performed by the processor 102, however, it should beunderstood that any suitable processor or processing circuitry mayperform the process 190 to perform a frequency determination operation.Furthermore, although blocks 192-208 are depicted in a particular order,it should be understood that the processor 102 may perform fewer oradditional operations in any suitable order when performing the process190. Additionally, the process 190 is described below generically asbased on recorded sensing data of a waveform of the power deliverysystem 20. It is noted that the below described method may be applied tocurrent waveforms and/or voltage waveforms when determining a frequencyof the power delivery system 20 and/or of the waveform.

At block 192, the processor 102 may receive a sampled waveform over timeas sensed data values. The sampled waveform may be discretely-sensedvoltage or current values recorded over time via sensing circuitrycoupled to the processor 102. The processor 102 may store the senseddata values into the memory 104, such as a part of a historic sensingdata record. The processor 102 may associate a time of sensing (e.g.,time stamp) with the sensed data values in the memory 104.

At block 194, the processor 102 may search for a previous value of thederivative ({dot over (v)}(t_(x))) at time t_(x) for which Equation 1 issatisfied:

$\begin{matrix}{{v( t_{0} )} = \frac{\overset{.}{v}( t_{x} )}{2\; \pi \frac{1}{4( {t_{0} - t_{x}} )}}} & \lbrack 1\rbrack\end{matrix}$

where t₀ is the time of the presently sensed value of the waveform andv(t₀) is the value of the presently sensed waveform. The derivative maybe found by any suitable method, such as by using historically recordedsensed voltage data or current data to determine the derivative of thewaveform. The processor 102 may set the frequency (F) to:

$\begin{matrix}{F = \frac{1}{4( {t_{0} - t_{x}} )}} & \lbrack 2\rbrack\end{matrix}$

Equivalently, the processor 102 may determine a scaled derivative ({dotover (V)}(t)) of the electrical waveform based on the computedderivative ({dot over (v)}(t)) and the time between the presently sensedvalue of the waveform and the time of the derivative sample, where thescaled derivative may be given as:

$\begin{matrix}{{\overset{.}{V}(t)} = \frac{\overset{.}{v}(t)}{2\; \pi \frac{1}{4( {t_{0} - t} )}}} & \lbrack 3\rbrack\end{matrix}$

The processor 102 may determine the time t_(x) corresponding to {dotover (V)}(t_(x)) being equal to the presently sensed value of theelectrical waveform. The processor 102 may determine the frequency basedon the time of the presently sensed value of the waveform and the timet_(x) as described in Equation 2.

To help illustrate, FIG. 6 is a graph 228 showing, generally, acomparison of the process 190. The graph 228 compares voltage amplitude(ordinate 230) over time (abscissa 172) of a voltage waveform (line 232)and of a derivative of the voltage waveform (line 234). It is noted thatthe derivative of the voltage waveform (line 234) is scaled by a factorequal to a for ease of plotting and may be scaled by an additionalamount to permit a fair comparison between the voltage waveform and thederivative of the voltage waveform. When the processor 102 applies therelationship defined via Equation 1 to a sensed waveform, the processor102 may compare a sensing value (v) sensed at the present time (t₀) todetermined derivative values of the sensed waveform at various previoustimes that are scaled according to the time difference between thepresent time and the times of the particular derivative values todetermine a frequency of the waveform before a period (e.g., frequencycycle) of the waveform completes. Graphically, when a prediction iscorrect, the prediction links the derivative of the voltage waveform(line 234) to the voltage waveform (line 234), as is shown with thevalue 238 but not the value 236. Using the derivative of the waveform todetermine a frequency of the waveform may improve calculation speeds offrequency determinations. For example, an efficiency of determining afrequency of a voltage waveform may improve (e.g., be increased,calculation made faster) since the frequency is able to be determined inless than a full cycle of the voltage waveform.

To help illustrate, FIG. 7 is another graph 250 of the comparisonoperation. The graph 250 compares voltage amplitude (ordinate 230) overtime (abscissa 172) of a voltage waveform (line 232) and of a derivativeof the voltage waveform (line 234). It is noted that, similar to thegraph 228, the derivative of the voltage (line 234) is scaled by afactor equal to 2π for ease of plotting. As described above, theprocessor 102 may apply Equation 1 and Equation 2 to determine thefrequency. As depicted in the graph 250, the processor 102 may comparethe voltage value sensed at the present time to and derivative valuescorresponding to previous sensed voltage values to determine which ofthe derivative values is to be used to determine the frequency of thevoltage.

In this example, the frequency corresponding to value 238 is correctbecause the predicted voltage (e.g., calculated expected voltage, v(t₀))equals the actual sensed voltage (e.g., falls on the same line).However, in this example, the frequencies corresponding to the value 236may be incorrect because the predicted voltages do not equal the actualsensed voltage, nor are within a threshold of the actual sensed voltage.While using the process 190, the processor 102 may thus retain and usethe frequency corresponding to the value 238 but discard and ignore thefrequencies corresponding to the values 236.

Returning to FIG. 5, at block 196, the processor 102 may determinewhether the time determined at block 194

$( {{e.g.},{t_{samp} = {t_{0} - \frac{1}{4F}}}} )$

is a time represented within the set of sampled values that define thewaveform. As discussed above, the waveform is sensed via many discretesampling operations resulting in multiple sampling values. The multiple,discrete sampling values are used to numerically determine thederivative of the waveform. In this way, when it is determined that aprevious sampled value corresponding to a scaled derivative of thewaveform at a first time equals a presently sampled value sensed at apresent time, the time determined is represented within the set ofsampled values that define the waveform. However, when nodiscretely-obtained derivative value corresponds to the presentlysampled value (e.g., the presently sampled value equals a voltagebetween two discrete derivative values scaled by a scaling factor ofEquation 1), the time determined is represented outside of the set ofsampled values that define the waveform.

To help illustrate, FIG. 8 is a graph 266 depicting a situation in whichthe time determined at block 194 may not be represented within the setof sampled values that define the waveform. The graph 266 comparesvoltage amplitude (ordinate 230) over time (abscissa 172) of a voltagewaveform (line 232) and of a derivative of the voltage waveform (line234). It is noted that, similar to the graph 228, the derivative of thevoltage waveform (line 234) is scaled by a factor equal to a for ease ofplotting. The graph 266 highlights how in some cases, the processor 102may determine that none of the frequency predictions yield a suitablecalculated expected voltage. In particular, the values 236,corresponding to a time shifted derivative of the waveform and scaled bythe scaling factor, show that the present sensed value is notrepresented within the set of sampled values that define the waveform.

Returning to FIG. 5, when the time determined at block 194 isrepresented within the set of sampled values that define the waveform,the processor 102, at block 198, may use the difference between the timedetermined at block 194 to determine the frequency (e.g., by applyingthe relationship defined in Equation 1). However, when the timedetermined at block 194 is determined to be between times of the set ofsampled values, the processor 102 performs an alternative method fordetermining the frequency (e.g., operations of blocks 204-210).

Thus, at block 204, the processor 102 may determine whether a differencebetween the waveform and the derivative of the waveform (or the scaledderivative of the waveform) has multiple zero crossings. The processor102 may subtract the derivative of the waveform from the waveform. Theprocessor 102 may then determine where the difference changes sign(e.g., a zero crossing) and may use sampled values corresponding to thezero crossings to determine the frequency. For example, the processor102 may determine flanking sampled values to the zero crossing (e.g.,when the zero crossing is at a N sample number between a sample set ofA, B, N, C, D, the flanking sampled values are B and C) and interpolatethe flanking sampled values to identify the frequency corresponding tothe time of zero crossing (e.g., perform an interpolation operation).That is, the processor 102 may determine a time t_(x) in which the valueof the scaled derivative is substantially equal to the presently sampledvalue of the waveform by interpolating the difference between values ofthe scaled derivative and the waveform.

When the determined difference between the waveform and the scaledderivative of the waveform does not include multiple zero crossings, theprocessor 102 may, at block 206, interpolate frequencies correspondingto two nearby sampled values of the waveform to determine the frequency.

To help illustrate, FIG. 9 is a graph 278 showing the difference betweeneach of the values 236 of the scaled derivative and the presentlysampled value of the signal (line 232) of the waveform applied to thesimulated voltages of the graph 266 of FIG. 8. The graph 278 comparesdifference amplitude (ordinate 280) over time (abscissa 282) in samplesfrom an initial time (e.g., t=0) of a voltage (line 232) and of aderivative of the voltage (line 234).

It is also noted that the resulting frequency predictions correspondingto each time are shown adjacent to the time axis for samples between t=0and t=−7. In this example, the processor 102 may determine that thedifference changes sign between the t=−6 samples and t=−7 samples, wherefrequency of the waveform at t=−6 samples corresponds to 1.3 Hz and thet=−7 samples corresponds to 1.1 Hz. This change happens at a zero axis284. Thus, to determine the frequency, the processor 102 may interpolatethe predicted frequencies for the t=−6 and t=−7 samples to determinethat the voltage has a frequency equal to 1.2 Hz.

The processor 102 may determine a first distance between the scaledderivative (e.g., second value 236A) at a third time (e.g., t=−6) andthe first value (e.g., presently sampled value of the signal (line232)). The process may continue by determining a second distance betweenthe scaled derivative (e.g., second value 236B) at a fourth time (e.g.,t=−7) and the first value (e.g., presently sampled value). The processor102 may then interpolate between the first distance and the seconddistance to obtain the zero crossing time (t_(x)) (e.g., a second timebetween times t=−6 and t=−7) at which the scaled derivativesubstantially equals (according to interpolation) the first value (e.g.,presently sampled value of the signal (line 232)). The processor 102 maydetermine the frequency as the time duration between the presentlysampled time (t₀) and the zero crossing time by using Equation 2. Whilethe interpolation is described herein as between two values (e.g.,linearly), any suitable interpolation or extrapolation may be used.

Returning to FIG. 5, in response to determining that the determineddifference between the waveform and the scaled derivative of thewaveform includes multiple zero crossings, the processor 102, at block208, selects the zero crossing based on the derivative of the determineddifference between the waveform and the derivative of the waveform. Inthis way, the processor 102 may analyze rates of change (e.g., slopes)associated with discretely sampled values of the waveform (and thevarious derivatives and differences of the waveform) to determine whichzero crossing may be used to determine the frequency. For example, whenthe most recently determined derivative of the difference (e.g., thedifference between the waveform and the scaled derivative of thewaveform) is positive, the processor 102 may choose the zero crossingcorresponding to a positive rate of change (e.g., positive slope, arelatively largest positive rate of change). However, when the mostrecently determined derivative of the difference (e.g., the differencebetween the waveform and the scaled derivative of the waveform) isnegative, the processor 102 may choose the zero crossing correspondingto a negative rate of change (e.g., negative slope, relatively largestnegative rate of change).

To help illustrate, FIG. 10A is a graph 296 depicting a simulatedwaveform that has multiple zero crossings of the zero axis 284. FIG. 10Bis a graph 300 depicting a simulated waveform that has multiple zerocrossings of the zero axis 284. For ease of description, FIG. 10A andFIG. 10B are explained together herein. The graph 296 comparesdifference amplitude (ordinate 280) over frequency (abscissa 298). Thegraph 300 also compares difference amplitudes (ordinate 280) overfrequency (abscissa 298).

Consider first the graph 296. The graph 296 has zeros 302 at about 7 Hz,11 Hz, 16 Hz, 60 Hz, and 112 Hz (e.g., where the difference changessign). The processor 102 may determine which of the zeros 302 correspondto a positive slope and that there are more than two zeros 302. Theselected zero of the graph 296 that is to correspond to the determinedfrequency may be 60 Hz. Furthermore, when considering the graph 300, thegraph 300 has zeros 302 at about 8 Hz, 10 Hz, 16 Hz, 60 Hz, and 120 Hz.In this case, the processor 102 may select the zero 302 corresponding tothe zero 302 having a negative slope. The selected zero 302 in thisexample also equals 60 Hz.

Returning to FIG. 5, once the zero crossing is selected, the processor102, may use the selected zero crossing in operations of block 206 todetermine the frequency of the waveform. For example, the processor 102may interpolate frequencies corresponding to nearby sampled values todetermine the frequency of the waveform.

At block 210, the processor 102 may use the frequency determined atblock 198 or block 206 to determine a magnitude and/or phase of thewaveform at the present time of sensing. The magnitude and/or phase ofthe waveform may be used to represent the waveform in a certain domain,such as a phasor domain. Any suitable method may be used to determine amagnitude and/or phase of the waveform.

At block 212, the processor 102 may use the frequency, magnitude, and/orphase in a control operation to adjust one or more operations of thepower delivery system 20. For example, the processor 102 may operate tocause transmission of a control signal to operate a circuit breaker(e.g., the circuit breaker 44, the circuit breaker 54) to close oroperate another protective device. A variety of control operations maybenefit from a frequency determination with improved efficiency. Asanother example, the processor 102 may use the frequency determinationwhen performing synchrophasing operations (e.g., determining when a loadside of an IED has a same frequency and/or phase as a line side of theIED).

In some cases, a magnitude of the waveform may change during a datawindow. When this happens, a ratio between magnitudes may be used todetermine the frequency of the waveform at a present time. The datawindow may refer to a number of data samples used for a calculation of aparticular quantity. In this case, the data window thus may be thenumber of data samples used to determine the frequency, or for example,a data window corresponding to the one-fourth cycle time interval (e.g.,to −1/(4F)).

For example, using a voltage waveform for the following example, bothEquation 4 and Equation 5 use the derivative of the voltage at thepresent time, {dot over (v)}(t₀), the determined frequency, F, and avoltage at the first time v(t₀) to determine the magnitude and phase ofthe voltage waveform. At the time determined at block 198, the magnitudemay be determined using the Equation 6.

$\begin{matrix}{m_{Fexact} = \sqrt{( \frac{\overset{.}{v}( t_{0} )}{2\; \pi \; F} )^{2} + ( {v( t_{0} )} )^{2}}} & \lbrack 4\rbrack \\{{phase} = {a\; \tan \; 2( {\frac{\overset{.}{v}( t_{0} )}{2\; \pi \; F},{v( t_{0} )}} )}} & \lbrack 5\rbrack \\{m_{F}\sqrt{( \frac{\overset{.}{v}( {t_{0} - \frac{1}{4F}} )}{2\; \pi \; F} )^{2} + ( {v( {t_{0} - \frac{1}{4F}} )} )^{2}}} & \lbrack 6\rbrack\end{matrix}$

The relationships of Equation 4 and Equation 6 may be used to determinethe frequency of the waveform. For example, Equation 8 uses thederivative of the voltage at the time determined at block 198

$( {\overset{.}{v}( {t_{0} - \frac{1}{4F}} )} ),$

a presently sensed value ({dot over (v)}(t₀)), the determined frequency(F), a voltage at the first time (v(t₀)), and a change in voltagemagnitude

$( \frac{m_{0F}}{m_{F}} )$

to determine the frequency. After the frequency is determined usingEquation 7, the magnitude and phase may be determined using Equation 4and Equation 5.

$\begin{matrix}{0 = {{\frac{m_{0F}}{m_{F}}\frac{\overset{.}{v}( {t_{0} - \frac{1}{4F}} )}{2\; \pi \; F}} = {v( t_{0} )}}} & \lbrack 7\rbrack\end{matrix}$

In some cases, the processor 102 may improve an accuracy of itsfrequency determination operations by at least in part applying theEquation 7 at local maximums. For example, the processor 102 may applythe Equation 7 when the absolute value of the voltage and the absolutevalue of the derivative of the voltage are both away from a zero (or athreshold amount away from a zero). In some cases, this is satisfiedaround 45 degrees (°), 135°, −135°, and −45° of the waveform, where thedegrees correspond to relatively defined points of a sinusoidal cyclefor the waveform.

In another example, the processor 102 may additionally or alternativelyrecover a value of the waveform at a present time by integrating of thederivative of the waveform from a time corresponding to a local maximumuntil the present time. The accuracy of the operations of the processor102 may improve if the processor 102 begins its integration of thederivative of the voltage at a zero crossing of the derivative of thevoltage, rather than a local maximum or minimum.

By using the system and methods described above, accuracy and generalperformance of the frequency determination operations may improve. Forexample, FIG. 11 is a graph 314 of the determined frequency of asimulated voltage waveform determined via the process 190 of FIG. 5. Thegraph 314 compares error magnitude (ordinate 316) over time in seconds(abscissa 318) for both a true frequency determination 320 (e.g.,error=0) and for a measured frequency determination 322 (e.g., non-zeroerror). In this example, a 61.5 Hz voltage was simulated formeasurement. It is noted that the steady state performance of thedescribed systems and methods when applied to this example may be higherthan 0.3 millihertz (mHz).

As an additional example, FIG. 12 is a graph 334 of an accuracy of thefrequency of the voltage determined via the process 190 of FIG. 5. Thegraph 334 compares frequency in Hz (ordinate 336) over time in seconds(abscissa 318) for both a true frequency determination 338 (e.g.,error=0) and for a measured frequency determination 340 (e.g., non-zeroerror). FIG. 12 highlights how the simulated performance of the systemsand methods described herein may yield relatively reliable frequencydeterminations.

Thus, technical effects of the present disclosure include systems andmethods for determining a frequency of a voltage and/or of a current ofa power delivery system based at least in part on derivativecalculations. By using the methods described above, a frequencydetermination system (e.g., frequency determination subsystem includinga processor and a memory of a controller) may obtain measurements withina quarter-cycle of the measured signals and with relatively lowprocessing burdens. Furthermore, these methods may permit the frequencydetermination system to obtain a frequency measurement for a relativelybroad range of frequencies (e.g., 1 Hz to 120 Hz). These improvedmethods may improve power delivery system operation by enabling controland/or protection circuitry to operate in a more efficient manner (e.g.,when the baseline frequency measurement is improved and/or moreefficient). This may improve a response of the power delivery system toa fault condition, or otherwise detected abnormal operation.

While specific embodiments and applications of the disclosure have beenillustrated and described, it is to be understood that the disclosure isnot limited to the precise configurations and components disclosedherein. For example, the systems and methods described herein may beapplied to an industrial electric power delivery system or an electricpower delivery system implemented in a boat or oil platform that may ormay not include long-distance transmission of high-voltage power.Accordingly, many changes may be made to the details of theabove-described embodiments without departing from the underlyingprinciples of this disclosure. The scope of the present disclosureshould, therefore, be determined only by the following claims.

The embodiments set forth in the present disclosure may be susceptibleto various modifications and alternative forms, specific embodimentshave been shown by way of example in the drawings and have beendescribed in detail herein. However, it may be understood that thedisclosure is not intended to be limited to the particular formsdisclosed. The disclosure is to cover all modifications, equivalents,and alternatives falling within the spirit and scope of the disclosureas defined by the following appended claims. In addition, the techniquespresented and claimed herein are referenced and applied to materialobjects and concrete examples of a practical nature that demonstrablyimprove the present technical field and, as such, are not abstract,intangible or purely theoretical. Further, if any claims appended to theend of this specification contain one or more elements designated as“means for [perform]ing [a function] . . . ” or “step for [perform]ing[a function] . . . ”, it is intended that such elements are to beinterpreted under 35 U.S.C. 112(f). For any claims containing elementsdesignated in any other manner, however, it is intended that suchelements are not to be interpreted under 35 U.S.C. 112(f).

What is claimed is:
 1. A system comprising: sensing circuitry configuredto periodically sense, at a defined interval, a value of an electricalwaveform on a transmission line of an electrical power transmissionsystem; and processing circuitry configured to determine a frequency ofthe electrical waveform at least in part by: receiving a plurality ofvalues of the electrical waveform sensed by the sensing circuitry,wherein the plurality of values comprises: a first value sensed by thesensing circuitry at a first time; and a plurality of previous valuessensed by the sensing circuitry at times before the first time;determining a scaled derivative of the electrical waveform based on oneor more of the plurality of values of the electrical waveform and a timeto the first time; determining a second time corresponding to a timebefore the first time at which the scaled derivative of the electricalwaveform has a value substantially equal to the first value of theelectrical waveform; and determining the frequency based at least inpart on a time duration between the second time and the first time;wherein the processing circuitry is configured to cause an operation ofthe electrical power transmission system to be controlled based at leastin part on the frequency.
 2. The system of claim 1, wherein theelectrical waveform comprises a voltage waveform.
 3. The system of claim1, wherein the electrical waveform comprises a current waveform.
 4. Thesystem of claim 1, wherein the processing circuitry is configured tooperate a protective device in response to the frequency being greaterthan a threshold.
 5. The system of claim 1, wherein the first valuerepresents a most recently sensed value of the electrical waveform. 6.The system of claim 1, wherein the processing circuitry is configuredtransmit the frequency to a supervisory control and data acquisition(SCADA) system.
 7. The system of claim 1, wherein the processingcircuitry is configured to: determine a first distance between thescaled derivative at a third time and the first value; determine asecond distance between the scaled derivative at a fourth time and thefirst value; and interpolate a zero crossing time as the second timeusing the third time and the fourth time from the first and seconddistance.
 8. The system of claim 7, wherein the interpolation compriseslinear interpolation between the third time and the fourth time.
 9. Thesystem of claim 7, wherein the processing circuitry is configured toselect the zero crossing time associated with a positive slope of adifference between the electrical waveform and the scaled derivative ofthe electrical waveform when a present derivative of the electricalwaveform is positive.
 10. The system of claim 1, wherein the processingcircuitry is configured to determine the scaled derivative as aderivative of the electrical waveform that is scaled by a scaling factorthat varies based on a difference between the first time and the secondtime.
 11. A method, comprising: receiving an input waveform comprising aplurality of sampled values each respectively sampled at time intervalsequal to a defined period, wherein a first sensed value is received at afirst time, and wherein the input waveform is a set of sampling valuesof an electrical waveform on a power line of an electrical powerdelivery system; determining a frequency of the input waveform at thefirst time based at least in part on a relationship between the inputwaveform and a derivative of the input waveform; and controlling anoperation of an electric power delivery system based at least in part onthe frequency of the input waveform.
 12. The method of claim 11,comprising: determining a second time corresponding to when thederivative of the input waveform is scaled by a scaling factor to equalthe first sensed value, wherein the scaling factor depends on adifference between the first time and the second time; and determiningthe frequency based at least in part on a time difference between thefirst time and the second time.
 13. The method of claim 11, comprising:determining whether the first sensed value is substantially equal to aninterpolated value between two calculated scaled derivative values fromthe derivative of the waveform; and determining the frequencycorresponding to two closest difference values of a difference betweenthe input waveform and the scaled derivative of the input waveformrelative to the zero crossing.
 14. The method of claim 11, comprisingdetermining changes in magnitude between previous values of the inputwaveform to determine the derivative of the input waveform.
 15. Themethod of claim 11, comprising determining a second time of a scaledderivative of the input waveform by: determining a first distancebetween the scaled derivative at a third time and the first value;determining a second distance between the scaled derivative at a fourthtime and the first value; and interpolating a zero crossing time as thesecond time based at least in part on the first distance and the seconddistance.
 16. The method of claim 15, wherein the processing circuitryis configured to: determine the frequency based on the zero crossingtime and the first time.
 17. A tangible, non-transitory,computer-readable medium comprising instructions that, when executed bya processor, cause the processor to: receive input waveform data sampledat a first time from sensing circuitry configured to periodically sense,at a defined interval, a value of an input waveform on a power line ofan electrical power delivery system; determine a second time to be atime at which a scaled derivative of the input waveform data at thesecond time substantially equals the input waveform data at the firsttime; determine a time difference between the first time and the secondtime; and calculate a frequency of the input waveform data based atleast in part on the time difference between the first time and thesecond time, wherein the frequency is calculated before a waveformcomprising the input waveform data completes a frequency cycle.
 18. Thetangible, non-transitory, computer-readable medium of claim 17,comprising instructions that cause the processor to: numericallygenerate intermediary derivative values based on data from the inputwaveform; and search, in the intermediary derivative values, for thesecond time at which the scaled derivative substantially equals theinput waveform data at the first time.
 19. The tangible, non-transitory,computer-readable medium of claim 17, comprising instructions that causethe processor to: calculate the time difference based at least in parton an interpolation operation.
 20. The tangible, non-transitory,computer-readable medium of claim 17, comprising instructions that causethe processor to calculate the derivative of the input waveform datawhen the input waveform data at the first time comprises a localmaximum.
 21. The tangible, non-transitory, computer-readable medium ofclaim 17, comprising instructions that cause the processor to calculatethe derivative of the input waveform data when the input waveform dataat the first time corresponds to 45°, 135°, −135°, or −45°.
 22. Thetangible, non-transitory, computer-readable medium of claim 17,comprising instructions that cause the processor to transmit thefrequency thereby adjusting operation of a component of an electricpower delivery system based at least in part on the frequency.